2 INTRODUCTORY MATHEMATICAL ANALYSIS 0. Review of AlgebraApplications and More AlgebraFunctions and GraphsLines, Parabolas, and SystemsExponential and Logarithmic FunctionsMathematics of FinanceMatrix AlgebraLinear ProgrammingIntroduction to Probability and Statistics
3 INTRODUCTORY MATHEMATICAL ANALYSIS Additional Topics in ProbabilityLimits and ContinuityDifferentiationAdditional Differentiation TopicsCurve SketchingIntegrationMethods and Applications of IntegrationContinuous Random VariablesMultivariable Calculus
4 Chapter 5: Mathematics of Finance Chapter ObjectivesTo solve interest problems which require logarithms.To solve problems involving the time value of money.To solve problems with interest is compounded continuously.To introduce the notions of ordinary annuities and annuities due.To learn how to amortize a loan and set up an amortization schedule.
5 Chapter Outline Compound Interest 5.1) Present Value Chapter 5: Mathematics of FinanceChapter OutlineCompound InterestPresent ValueInterest Compounded ContinuouslyAnnuitiesAmortization of Loans5.1)5.2)5.3)5.4)5.5)
6 Chapter 5: Mathematics of Finance 5.1 Compound InterestExample 1 – Compound InterestCompound amount S at the end of n interest periods at the periodic rate of r is asSuppose that $500 amounted to $ in a savings account after three years. If interest was compounded semiannually, find the nominal rate of interest, compounded semiannually, that was earned by the money.
7 There are 2 × 3 = 6 interest periods. Chapter 5: Mathematics of Finance5.1 Compound InterestExample 1 – Compound InterestSolution:There are 2 × 3 = 6 interest periods.The semiannual rate was 2.75%, so the nominal rate was 5.5 % compounded semiannually.
8 The periodic rate is r = 0.06/4 = 0.015. Chapter 5: Mathematics of Finance5.1 Compound InterestExample 3 – Compound InterestHow long will it take for $600 to amount to $900 at an annual rate of 6% compounded quarterly?Solution:The periodic rate is r = 0.06/4 =It will take
9 Example 5 – Effective Rate Effective Rate Chapter 5: Mathematics of Finance5.1 Compound InterestExample 5 – Effective RateEffective RateThe effective rate re for a year is given byTo what amount will $12,000 accumulate in 15 years if it is invested at an effective rate of 5%?Solution:
10 Respective effective rates of interest are Chapter 5: Mathematics of Finance5.1 Compound InterestExample 7 – Comparing Interest RatesIf an investor has a choice of investing money at 6% compounded daily or % compounded quarterly, which is the better choice?Solution:Respective effective rates of interest areThe 2nd choice gives a higher effective rate.
11 Chapter 5: Mathematics of Finance 5.2 Present ValueExample 1 – Present ValueP that must be invested at r for n interest periods so that the present value, S is given byFind the present value of $1000 due after three years if the interest rate is 9% compounded monthly.Solution:For interest rate,Principle value is
12 a final payment at the end of five years. Chapter 5: Mathematics of Finance5.2 Present ValueExample 3 – Equation of ValueA debt of $3000 due six years from now is instead to be paid off by three payments:$500 now,$1500 in three years, anda final payment at the end of five years.What would this payment be if an interest rate of 6% compounded annually is assumed?
13 The equation of value is Chapter 5: Mathematics of Finance5.2 Present ValueSolution:The equation of value is
14 Example 5 – Net Present Value Chapter 5: Mathematics of Finance5.2 Present ValueExample 5 – Net Present ValueNet Present ValueYou can invest $20,000 in a business that guarantees you cash flows at the end of years 2, 3, and 5 as indicated in the table.Assume an interest rate of 7% compounded annually and find the net present value of the cash flows.YearCash Flow2$10,0003800056000
15 Solution: Chapter 5: Mathematics of Finance 5.2 Present Value Example 5 – Net Present ValueSolution:
16 5.3 Interest Compounded Continuously Chapter 5: Mathematics of Finance5.3 Interest Compounded ContinuouslyExample 1 – Compound AmountCompound Amount under Continuous InterestThe compound amount S is defined asIf $100 is invested at an annual rate of 5% compounded continuously, find the compound amount at the end ofa. 1 year.b. 5 years.
17 Effective Rate under Continuous Interest Chapter 5: Mathematics of Finance5.3 Interest Compounded ContinuouslyEffective Rate under Continuous InterestEffective rate with annual r compounded continuously isPresent Value under Continuous InterestPresent value P at the end of t years at an annual r compounded continuously is .
18 We want the present value of $25,000 due in 20 years. Chapter 5: Mathematics of Finance5.3 Interest Compounded ContinuouslyExample 3 – Trust FundA trust fund is being set up by a single payment so that at the end of 20 years there will be $25,000 in the fund. If interest compounded continuously at an annual rate of 7%, how much money should be paid into the fund initially?Solution:We want the present value of $25,000 due in 20 years.
19 5.4 Annuities Sequences and Geometric Series Chapter 5: Mathematics of Finance5.4 AnnuitiesExample 1 – Geometric SequencesSequences and Geometric SeriesA geometric sequence with first term a and common ratio r is defined asa. The geometric sequence with a = 3, common ratio 1/2 , and n = 5 is
20 b. Geometric sequence with a = 1, r = 0.1, and n = 4. Chapter 5: Mathematics of Finance5.4 AnnuitiesExample 1 – Geometric Sequencesb. Geometric sequence with a = 1, r = 0.1, andn = 4.c. Geometric sequence with a = Pe−kI , r = e−kI ,n = d.Sum of Geometric SeriesThe sum of a geometric series of n terms, with first term a, is given by
21 Find the sum of the geometric series: Chapter 5: Mathematics of Finance5.4 AnnuitiesExample 3 – Sum of Geometric SeriesFind the sum of the geometric series:Solution: For a = 1, r = 1/2, and n = 7Present Value of an AnnuityThe present value of an annuity (A) is the sum of the present values of all the payments.
22 Chapter 5: Mathematics of Finance 5.4 AnnuitiesExample 5 – Present Value of AnnuityFind the present value of an annuity of $100 per month for years at an interest rate of 6% compounded monthly.Solution:For R = 100, r = 0.06/12 = 0.005, n = ( )(12) = 42From Appendix B, .Hence,
23 Chapter 5: Mathematics of Finance 5.4 AnnuitiesExample 7 – Periodic Payment of AnnuityIf $10,000 is used to purchase an annuity consisting of equal payments at the end of each year for the next four years and the interest rate is 6% compounded annually, find the amount of each payment.Solution:For A= $10,000, n = 4, r = 0.06,
24 The amount S of ordinary annuity of R for n periods at r per period is Chapter 5: Mathematics of Finance5.4 AnnuitiesExample 9 – Amount of AnnuityAmount of an AnnuityThe amount S of ordinary annuity of R for n periods at r per period isFind S consisting of payments of $50 at the end of every 3 months for 3 years at 6% compounded quarterly. Also, find the compound interest.Solution: For R=50, n=4(3)=12, r=0.06/4=0.015,
25 Chapter 5: Mathematics of Finance 5.4 AnnuitiesExample 11 – Sinking FundA sinking fund is a fund into which periodic payments are made in order to satisfy a future obligation. A machine costing $7000 is replaced at the end of 8 years, at which time it will have a salvage value of $700. A sinking fund is set up. The amount in the fund at the end of 8 years is to be the difference between the replacement cost and the salvage value. If equal payments are placed in the fund at the end of each quarter and the fund earns 8% compounded quarterly, what should each payment be?
26 Amount needed after 8 years = 7000 − 700 = $6300. Chapter 5: Mathematics of Finance5.4 AnnuitiesExample 11 – Sinking FundSolution:Amount needed after 8 years = 7000 − 700 = $6300.For n = 4(8) = 32, r = 0.08/4 = 0.02, and S = 6300, the periodic payment R of an annuity is
27 5.5 Amortization of Loans Amortization Formulas Chapter 5: Mathematics of Finance5.5 Amortization of LoansAmortization Formulas
28 total interest charges, and principal remaining after five years. Chapter 5: Mathematics of Finance5.5 Amortization of LoansExample 1 – Amortizing a LoanA person amortizes a loan of $170,000 by obtaining a 20-year mortgage at 7.5% compounded monthly. Findmonthly payment,total interest charges, andprincipal remaining after five years.
29 Solution: a. Monthly payment: b. Total interest charge: Chapter 5: Mathematics of Finance5.5 Amortization of LoansExample 1 – Amortizing a LoanSolution:a. Monthly payment:b. Total interest charge:c. Principal value: